Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrödinger equation

In this paper, we propose new efficient and accurate numerical methods for computing dark solitons and review some existing numerical methods for bright and/or dark solitons in the nonlinear Schrödinger equation (NLSE), and compare them numerically in terms of accuracy and efficiency. We begin with a review of dark and bright solitons of NLSE with defocusing and focusing cubic nonlinearities, respectively. For computing dark solitons, to overcome the nonzero and/or non-rest (or highly oscillatory) phase background at far field, we design efficient and accurate numerical methods based on accurate and simple artificial boundary conditions or a proper transformation to rest the highly oscillatory phase background. Stability and conservation laws of these numerical methods are analyzed. For computing interactions between dark and bright solitons, we compare the efficiency and accuracy of the above numerical methods and different existing numerical methods for computing bright solitons of NLSE, and identify the most efficient and accurate numerical methods for computing dark and bright solitons as well as their interactions in NLSE. These numerical methods are applied to study numerically the stability and interactions of dark and bright solitons in NLSE. Finally, they are extended to solve NLSE with general nonlinearity and/or external potential and coupled NLSEs with vector solitons.